I need some help with this question here, I'll explain why in a second. The question is:
$f(x,y)$ is a homogeneous function with order $3$. It is known that:
$f_{x}(2,1)=7$
and
$f_{y}(8,4)=5$
I need to find $f(12,6)$. The final answer should be (if there is no mistake in the book): $1030.5$.
I know the definition of homogeneous function, and I am aware of Euler's rule, but the point $(x,y)$ for $f$, $f_x$, and $f_y$, is not the same, that confuses me. I am also aware of a law saying that if $f$ is homogeneous then so is $f_x$ and $f_y$, with an order $2$. But that doesn't help me either. Can you assist please? My intuition say that there is some equation here that I am missing.
Thank you.
P.S.: Apologies for my LaTeX not working, I used an online editor, have no clue why it is not working, the LaTeX equations are the partial derivatives.
By Euler's homogeneous function theorem you have $$ f(12,6)=\frac{1}{3}(12,6)\cdot\nabla f(12,6)=\\ \quad=\frac{1}{3}\left(12\cdot f_x(12,6)+6\cdot f_y(12,6))\right)=\\ \quad=\frac{1}{3}\left(12\cdot6^2\cdot f_x(2,1)+6\cdot (3/2)^2\cdot f_y(8,4))\right) $$